Suppose you’re a flood response manager and you’re responsible for initiating a response if and when a flood event is forecast to occur. If the hydrologic forecast you receive is a deterministic, ‘single value’ forecast, the decision is relatively straightforward. You initiate the response if the predicted water level exceeds a critical threshold. There is very little else you can do, as you have no information on the (un-)certainty of the prediction.

Given this probability forecast, would you issue a warning?

If you receive a probabilistic forecast on the other hand, then how do you decide whether to take action? For example, if the prediction says that there is a 20% probability of flooding, what would you do? Sit still? Move? What if the predicted probability was 80%?

Your decision constitutes a choice between two alternatives: take action or don’t. Action comes at a cost C and may reduce damage to that which is unavoidable (Lu). If you don’t take action, no response costs are incurred but if a flood hits, damage will be the sum of unavoidable damage and avoidable damage (La + Lu).

Independent of your decision, the flood may or may not occur, with flood occurrence having some probability P which we assume is predicted correctly by your hydrologist.

The expected values of the action (E1) and no action (E0) decisions are the sum of investments in response costs, and the probability weighted flood damage:E1 = C + P*Lu
E0 = P*(La + Lu)

Assuming you’d want to base your decision on expected value only, you would only issue a warning if the expected value of ‘action’ is lower than that of ‘no action’, i.e. if E1 < E0

In words, you would only issue a warning if the predicted probability of flooding (P) is higher than the ratio of the cost of warning response (C) to the amount of damage that will be reduced by that action (La). This ratio is often referred to as the cost-to-loss ratio r. The optimal decision rule then is: warn if P > r.